1
GATE CSE 2014 Set 1
+2
-0.6
A function $$f(x)$$ is continuous in the interval $$\left[ {0,2} \right]$$. It is known that $$f(0)$$ $$=$$ $$f(2)$$ $$= -1$$ and $$f(1)$$ $$= 1$$. Which one of the following statements must be true?
A
There exists $$a$$ $$y$$ in the interval $$(0, 1)$$ such that $$f(y) =$$ $$f(y+1)$$
B
For every $$y$$ in the interval $$(0, 1)$$, $$f(y)$$ $$=$$ $$f(2 - y)$$
C
The maximum value of the function in the interval $$(0,2)$$ is $$1$$
D
There exists $$a$$ $$y$$ in the interval $$(0,1)$$ such that $$f(y)$$ $$=-$$$$f(2-y)$$
2
GATE CSE 2014 Set 1
Numerical
+2
-0
The function $$f(x) =$$ $$x$$ $$sinx$$ satisfies the following equation:
$$f$$"$$\left( x \right) + f\left( x \right) + t\,\cos \,x\,\, = \,\,0$$. The value of $$t$$ is ______ .
3
GATE CSE 2014 Set 3
+2
-0.6
The value of the integral given below is $$\int_0^\pi {{x^2}\,\cos \,x\,dx}$$\$
A
$$- 2\pi$$
B
$$\pi$$
C
$$-\pi$$
D
$$2\pi$$
4
GATE CSE 2014 Set 3
Numerical
+2
-0
Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre specified pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1+ min(T(y),T(z)). Then the value of the product yz is _______.